Geometry and Number Theory MPhil/PhD

London, Bloomsbury

The London School of Geometry and Number Theory (LSGNT) is an EPSRC-funded Centre for Doctoral Training (CDT) and is a joint venture between UCL, King's College London and Imperial College London. It offers a wide range of research projects in different aspects of number theory, geometry and topology.

UK students International students
Study mode
UK tuition fees (2023/24)
Overseas tuition fees (2023/24)
4 academic years
Programme starts
September 2023
Applications accepted
All applicants: 01 Nov – 31 Dec 2022

Applications closed

Entry requirements

Applicants are expected to have completed at least a very good four-year degree in mathematics or a closely related discipline by September 2020; in Europe this implies a Master’s degree for most applicants. We also welcome applications from outstanding US bachelor students, for example. Please note we do not expect you to have completed your four-year degree before you apply for our programme, but you must expect to complete this before your entry to our programme in September.

The English language level for this programme is: Level 1

UCL Pre-Master's and Pre-sessional English courses are for international students who are aiming to study for a postgraduate degree at UCL. The courses will develop your academic English and academic skills required to succeed at postgraduate level. International Preparation Courses

Further information can be found on our English language requirements page.

If you are intending to apply for a time-limited visa to complete your UCL studies (e.g., Student visa, Skilled worker visa, PBS dependant visa etc.) you may be required to obtain ATAS clearance. This will be confirmed to you if you obtain an offer of a place.

Equivalent qualifications

Country-specific information, including details of when UCL representatives are visiting your part of the world, can be obtained from the International Students website.

International applicants can find out the equivalent qualification for their country by selecting from the list below. Please note that the equivalency will correspond to the broad UK degree classification stated on this page (e.g. upper second-class). Where a specific overall percentage is required in the UK qualification, the international equivalency will be higher than that stated below. Please contact Graduate Admissions should you require further advice.

About this degree

Mathematics research at UCL is flourishing with significant funding from EPSRC, ERC, the Royal Society and industry. The department is a member of the UCL Institute of Origins, Centre for Inverse Problems, and CoMPLEX (Centre of Mathematics and Physics in the Life Sciences and Experimental Biology); it hosts the Clinical Operational Research Unit and has 20 postdoctoral researchers including the Clifford Fellow. The London School of Geometry and Number Theory is run by UCL in partnership with King's College London and Imperial College London. Students benefit from our leading role in the inter-university London Taught Course Centre for PhD Students in the Mathematical Sciences.

Who this course is for

Our research degree programme is aimed at graduate students in mathematics or related sciences. Applicants will have analytical skills, and have formed a strong interest in pursuing pure mathematical research.

What this course will give you

The training that we offer will differ from a traditional UK PhD in pure mathematics in a number of respects, and particularly in the first-year mathematical training that will take place across the three institutions. This training is designed to encourage the development of an original approach to problem solving that is attractive to both the academic and industrial sector. Our industrial partners have also had input on some aspects of our programme to reflect their industry requirements. The LSGNT has an ethos of working and learning together in a friendly and supportive environment. The collaborative spirit established in the first year will be maintained by a number of mechanisms including cross-college supervision teams, junior seminars, working groups, and other activities such as a new LSGNT colloquium, and intensive workshops. Our programme is set up to create a network for our students, so that they can collaborate and learn from each other throughout their four years and beyond. This structure is designed to provide a more supportive experience for the theoretical Mathematics PhD student.

The foundation of your career

Many students go on to careers in finance,  innovation, technology and research and development. Many others proceed to postdoctoral research and academic careers. The LSGNT CDT has several industrial partners including Digital Catapult, The Heilbronn Institute for Mathematical Research, the Bank of England, Morgan Stanley and G-Research.


The LSGNT CDT has now had three intakes of students who have graduated from our programme. All have found employment after attaining their 4 year research PhD with us. LSGNT Mathematics graduates have found positions in the academic field, at universities such as Princeton, Sorbonne Paris North, Max Planck Institute, Bonn, Leiden, Harvard, Institute for Advanced Studies, ETH Zurich etc.

Others have entered industry in sectors such as finance, technology startups and research and development. With companies such as Morgan Stanley, Eigen Technologies, Goldman Sachs, Lloyds, Credit Suisse, Quantica Capital and Huawei Technologies Research & Development (UK) Ltd.


Students interact closely with their peers from all three institutions within the CDT (UCL, Imperial and King’s) and those universities within London via the LTCC, as well as other UK universities including Bath, Cambridge and Oxford. Students attend seminars both within and outside the departments of all three institutions and are exposed to a vigorous conference programme. Students also interact with the CDT’s industrial partners, the Bank of England, Morgan Stanley, The Heilbronn Institute for Mathematical Research, Digital Catapult and G-Research.

Teaching and learning

In the first year a broad mathematical training is offered across the three participating institutions. The emphasis is on topics in geometry and number theory and will consist of topics modules and interface lectures, a computer programming course, seminars, lectures from both the London Taught Course Centre (LTCC) and the Taught Course Centre and two mathematical mini projects of between 15,000 and 50,000 words each.

Towards the end of their first year, Home/EU students will choose a project and supervisor from one of the two institutions indicated on their original offer, and enrol at the college of their primary supervisor where they will complete their mathematical research thesis of 100,000 words. Overseas students will have already been assigned to a supervisor/institution during their recruitment and will move to that institution at the start of year two.  All LSGNT CDT students will continue to have access to, and attend events at, all three institutions (UCL, King's and Imperial) during the entirety of their PhD programme.

In the third or fourth year, students will have the opportunity to spend up to three months either on an industrial placement or in a mathematics department outside London. The Heilbronn studentship gives the student the opportunity to spend two months per year during their PhD working at the Heilbronn Institute for Mathematical Research.

Students must complete the first year assessment outlined in the LSGNT course handbook, in order to pass year one and progress to their PhD research project in year two. This includes peer assessment, graded mini research projects and presentations and course attendance. Please note that this assessment does not include exams.

Contact Hours Year 1: Approximately 20-25 hours dependent on the optional courses selected and seminars attended.

Contact Hours Years 2-4: This will be dependent on the arrangement between the student and their PhD supervisor.

Research areas and structure

  • Geometry and Number Theory
  • Pure Mathematics

Research environment

Our research covers very large areas of modern geometry and number theory. In geometry, we cover the spectrum from algebraic geometry, complex geometry, symplectic geometry, through to differential and riemannian geometry, geometric analysis, including techniques from nonlinear elliptic and parabolic partial differential equations. Subjects such as geometric group theory, knot theory, discrete systems and inverse problems are also represented.

Number theory is similarly broad, with researchers in both analytic and algebraic number theory, with specialists in (p-adic) Galois representations, modular and automorphic forms, arithmetic geometry, arithmetic aspects of spectral theory, multiplicative functions and additive combinatorics.

Students attending our course spend their first year based at UCL and attend taught courses across all three institutions (UCL, Imperial and King's), as well as seminars and study groups.  The LSGNT offers a variety of lecture courses for students to choose from, as well as a core group of courses that they will be required to attend, including a computing course designed for pure mathematicians.  Students must also complete two mini projects with different supervisors before completing their first year.  Students will also attend a two day induction event outside of UCL including a professional skills course, such as presentation skills and will attend conferences/seminars in the UK and abroad relevant to the topic area they ultimately choose.

Year one assessment is outlined in the LSGNT coursebook.

Students do not have to decide on their final thesis project topic area until the end of their first year, when they will move to one of the two institutions that were listed in their offer letter.  At this point they will also confirm their supervisor.  Please note the exception to this is overseas students, who are assigned an institution at the time of their offer, though they may pick their supervisor and topic later.  Students continue to have access to all three institutions during the course of their 4 year PhD programme.


Details of the accessibility of UCL buildings can be obtained from AccessAble Further information can also be obtained from the UCL Student Support & Wellbeing team.

Fees and funding

Fees for this course

UK students International students
Fee description Full-time
Tuition fees (2023/24) £5,860
Tuition fees (2023/24) £26,200

The tuition fees shown are for the year indicated above. Fees for subsequent years may increase or otherwise vary. Where the programme is offered on a flexible/modular basis, fees are charged pro-rata to the appropriate full-time Master's fee taken in an academic session. Further information on fee status, fee increases and the fee schedule can be viewed on the UCL Students website:

Additional costs

Please note this studentship does not include funding for visa application fees or the NHS surcharge for overseas students.

This studentship includes funding to cover tuition fees, a tax free stipend for living costs, a laptop (provided by the universities) and a research support training grant for travel/fees/accommodation and meals to attend conferences/workshops/summer-winter schools held in the UK and the rest of the world.

Please note, disabled students may also apply for the disabled students allowance. Please visit the UCL Support and Wellbeing webpage for more information regarding this.

For more information on additional costs for prospective students please go to our estimated cost of essential expenditure at Accommodation and living costs.

Funding your studies

Our LSGNT studentships are all fully funded, including fees, a tax free stipend, a grant to cover travel expenses to attend conferences/workshops and a laptop. For 2022 we have funding for a total of 16 studentships (UK and International), funded by the EPSRC, UCL, Imperial College London, King's College London and our industrial partner, the Heilbronn Institute for Mathematical Research.

For a comprehensive list of the funding opportunities available at UCL, including funding relevant to your nationality, please visit the Scholarships and Funding website.

CSC-UCL Joint Research Scholarship

Value: Fees, maintenance and travel (Duration of programme)
Criteria Based on academic merit
Eligibility: EU, Overseas

Next steps

The EPSRC Centre for Doctoral Training in Geometry and Number Theory at the Interface: London School of Geometry and Number Theory was one of 75 Centres for Doctoral Training announced by UKRI to develop the skills needed for UK prosperity. Find out more:

Please note that you may submit applications for a maximum of two graduate programmes (or one application for the Law LLM) in any application cycle.

Got questions? Get in touch

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